## Stability of a functional equation for square root spirals.(English)Zbl 1016.39020

The authors study a concrete case of the stability (in the sense of Hyers-Ulam-Rassias) of the equation of the square root spiral: $f( \sqrt {r^2 +1}) = f (r) + \arctan \tfrac 1r.\tag{1}$ The main result is the following. If $$f: [1, \infty) \to [0, \infty)$$ satisfies, for all $$r \geq 1$$, $$|f (\sqrt {r^2 +1})- f(r)- \arctan {1 \over r} |$$ $$\leq g(r)$$, where $$g: [1 , \infty) \to [0, \infty)$$ is such that $$G(r) : = \sum_{k=0}^{\infty} g (\sqrt {r^2 + k}) < \infty$$, then there exists a unique solution $$F$$ of (1) such that, for all $$r \geq 1$$, $|F(r)- f(r) |\leq G(r).$

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations
Full Text:

### References:

  Heuvers, K.J.; Moak, D.S.; Boursaw, B., The functional equation of the square root spiral, (), 111-117 · Zbl 0976.39018  Ulam, S.M., Problems in modern mathematics, (1964), Wiley New York, Chapter VI · Zbl 0137.24201  Hyers, D.H., On the stability of the linear functional equation, Proc. nat. acad. sci. U.S.A., 27, 222-224, (1941) · Zbl 0061.26403  Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040  Găvruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043  Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, (1998), Birkhäuser Boston, MA · Zbl 0894.39012  Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, (2001), Hadronic FL · Zbl 0980.39024  Forti, G.L., Hyers-Ulam stability of functional equations in several variables, Aequationes math., 50, 143-190, (1995) · Zbl 0836.39007  Hyers, D.H.; Rassias, Th.M., Approximate homomorphisms, Aequationes math., 44, 125-153, (1992) · Zbl 0806.47056  Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations, Dynam. systems appl., 6, 541-566, (1997) · Zbl 0891.39025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.